Simplify and expand the following expression: $ \dfrac{7}{a + 6}-\dfrac{2a}{a - 7} $
Answer: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(a + 6)(a - 7)$ Multiply the first term by $\dfrac{a - 7}{a - 7}$ $ \begin{align*} \dfrac{7}{a + 6} \times \dfrac{a - 7}{a - 7} & = \dfrac{(7)(a - 7)}{(a + 6)(a - 7)} \\ & = \dfrac{7a - 49}{(a + 6)(a - 7)}\end{align*} $ Multiply the second term by $\dfrac{a + 6}{a + 6}$ $ \begin{align*} \dfrac{2a}{a - 7} \times \dfrac{a + 6}{a + 6} & = \dfrac{(2a)(a + 6)}{(a - 7)(a + 6)} \\ & = \dfrac{2a^2 + 12a}{(a - 7)(a + 6)}\end{align*} $ Now we have: $ = \dfrac{7a - 49}{(a + 6)(a - 7)} - \dfrac{2a^2 + 12a}{(a - 7)(a + 6)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{7a - 49 - (2a^2 + 12a)}{(a + 6)(a - 7)} $ $ = \dfrac{7a - 49 - 2a^2 - 12a}{(a + 6)(a - 7)} $ $ = \dfrac{-5a - 49 - 2a^2}{(a + 6)(a - 7)}$ Expand the denominator: $ = \dfrac{-5a - 49 - 2a^2}{a^2 - a - 42}$